We have that $\lceil x \rceil - \lfloor x \rfloor = 0.$ Then, what is $\lceil x \rceil - x$?
Answer: Given that $\lceil x \rceil - \lfloor x \rfloor = 0,$ we see that $x$ must be an integer. Otherwise, the ceiling of $x$ would be greater than the floor of $x.$ Therefore, $\lceil x \rceil = x$ and $\lceil x \rceil - x = \boxed{0}.$